An introduction to Lie groups and the geometry of homogeneous spaces / Andreas Arvanitoyeorgos.

Format:

Book

Author/Creator:

Arvanitogeōrgos, Andreas, 1963-

Series:

Student mathematical library 1520-9121 ; v. 22.

Student mathematical library, 1520-9121 ; v. 22

Standardized Title:

Homades Lie, homogeneis chōroi kai diaphorikē geōmetria. English

Language:

English

Greek, Modern (1453-)

Subjects (All):

Lie groups.

Homogeneous spaces.

Physical Description:

xvi, 141 pages : illustrations ; 22 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2003]

Contents:

Chapter 1. Lie Groups 1

2. Smooth manifolds: A review 2

3. Lie groups 8

4. The tangent space of a Lie group - Lie algebras 12

5. One-parameter subgroups 15

6. The Campbell-Baker-Hausdorff formula 20

7. Lie's theorems 21

Chapter 2. Maximal Tori and the Classification Theorem 23

1. Representation theory: elementary concepts 24

2. The adjoint representation 28

3. The Killing form 32

4. Maximal tori 36

5. The classification of compact and connected Lie groups 39

6. Complex semisimple Lie algebras 41

Chapter 3. The Geometry of a Compact Lie Group 51

1. Riemannian manifolds: A review 51

2. Left-invariant and bi-invariant metrics 59

3. Geometrical aspects of a compact Lie group 61

Chapter 4. Homogeneous Spaces 65

1. Coset manifolds 65

2. Reductive homogeneous spaces 71

3. The isotropy representation 72

Chapter 5. The Geometry of a Reductive Homogeneous Space 77

1. G-invariant metrics 77

2. The Riemannian connection 79

3. Curvature 80

Chapter 6. Symmetric Spaces 87

2. The structure of a symmetric space 88

3. The geometry of a symmetric space 91

4. Duality 92

Chapter 7. Generalized Flag Manifolds 95

2. Generalized flag manifolds as adjoint orbits 96

3. Lie theoretic description of a generalized flag manifold 98

4. Painted Dynkin diagrams 98

5. T-roots and the isotropy representation 100

6. G-invariant Riemannian metrics 103

7. G-invariant complex structures and Kahler metrics 105

8. G-invariant Kahler-Einstein metrics 108

9. Generalized flag manifolds as complex manifolds 111

Chapter 8. Advanced topics 113

1. Einstein metrics on homogeneous spaces 113

2. Homogeneous spaces in symplectic geometry 118

3. Homogeneous geodesics in homogeneous spaces 123.

Notes:

Includes bibliographical references (pages 129-137) and index.

ISBN:

0821827782

OCLC:

52980839